On Wed, Jan 18, 2006 at 05:40:56PM -0500, Cory Casanave wrote:
> A theoretical set of questions getting down to some practical
> realities...
>
> We have used the phrase "an ontology" and "a theory" quite a bit - do
> we really know what these are? (01)
Yes. :-) (02)
Ontologies will be most useful to us insofar as they are moved around on
the web and processed by automated reasoners. So I prefer a nice,
simple definition: (03)
An ontology is a set of sentences in a formal language. (04)
The sentences in the ontology are its *axioms*. The language of an
ontology must have a rigorous semantics that provides general rules for
interpreting the complex expressions of the language in terms of the
interpretations of its lexical primitives. Typically, of course, an
ontology will have an *intended interpretation*, as with, say, an
ontology of the natural numbers like Peano Arithmetic, or an ontology of
wine or a genomic ontology. But the ontology itself is nothing more or
less than the sentences it comprises. (05)
> Is "an ontology" and "a theory" the same? (06)
That depends. "Theory" is often defined simply to be a set of sentences
(e.g., by Enderton in his well known text), but equally often theories
are required to be deductively closed. So "Yes" if by "theory" we mean
the former sense, "No" (in general) if we mean it in the latter sense. (07)
> What does it mean for statements to be in one ontology Vs. Another? (08)
Sentence S is *in* ontology O iff S is a member of O. (09)
> What is the scope of an ontology? (010)
I don't know what "scope" means. (011)
> Do ontologies have identity? (012)
Yes. O1 = O2 iff the contain the same sentences. Note, of course, that
different ontologies can have exactly the same consequences. Such
ontologies are logically equivalent. There are weaker notions of
equivalence as well, e.g., if each theory can be "interpreted" in the
other, in a well understood sense. Many other very useful logical
relations between ontologies can be rigorously defined on this approach. (013)
> Do they change over time? (014)
No. Though of course one can trace change through an evolving series of
ontologies. (015)
> Do statements have identity? (016)
Yes. Each sentence in a formal language is constructed uniquely from
the lexical primitives by a unique composition of syntactic rules. (017)
> Is "an ontology" equivalent to "a file" of statements? (018)
No, files are concrete. The same ontology can reside in many files on
many computers. (019)
> How big is an ontology? How small can it be? (020)
Useful ontologies are expressed in countable languages and hence are
themselves countable in size, i.e., either finite or denumerably
infinite. The smallest ontology is the empty set. (021)
> If it is a lattice, what are the relations between the nodes? (022)
Ontologies are not lattices on this approach. (023)
> From what I can tell, the concept of "an ontology" has grown out of
> the practice of representing bits of knowledge in files. The ontology
> is a thing you can put in a "file" (Physical or logical). (024)
Agreed. (025)
> Perhaps the "semantics" of "an ontology" is that it is the speech act
> of a particular person at a particular time asserting an arbitrary
> collection of statements. (026)
On the proposed view, the semantics of an ontology would be specified by
a corresponding model theory that provides general rules for
interpreting the language of the ontology, together perhaps with a
specification of a particular intended model. (027)
> In other words, there is an act of publishing an ontology or theory. (028)
Ontologies are particularly useful when they are made available on the
web. I'm not sure how much importance should be accorded to the "act"
of doing so, beyond the fact that that is what makes the ontology
available, hence useful. (029)
> So is an ontology anything more than an arbitrary set of statements? (030)
In my view, no. Though of course, the axioms of useful ontology will be
very carefully chosen. (031)
> Is it more or less than a file? (032)
It is "other" than a file. A file "contains" an ontology if, properly
rendered to a screen or other output device, its content consists of the
sentences of that ontology. (033)
> Does it exist (034)
Sure, in whatever sense any abstract object does. (035)
> and have meaning? (036)
Sure, if supplied with a proper semantics. (037)
> Is it the semantic web? (038)
No. There will be many ontologies on the semantic web in the real
world. (Though I guess that in the semantic web in Plato's Heaven every
ontology is written in OWL and their union is consistent, so in that
case, yes, the semantic web is an ontology. :-) (039)
I spell out this viewpoint in a bit more detail in the Dagstuhl paper
noted in my previous post. For a good example of an ontology that
comports with this viewpoint, see the Process Specification Language
(PSL): (040)
http://www.mel.nist.gov/psl/ontology.html (041)
Chris Menzel (042)
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